poly2rc (Signal Processing Toolbox)

Conversion of prediction polynomial to reflection coefficients.

Syntax

k = poly2rc(a)
[k,r0] = poly2rc(a,efinal)

Description

k = poly2rc(a)

finds the reflection coefficients of the AR lattice structure corresponding to prediction polynomial a. a must be real, and a(1) cannot be 0. If a(1) is not equal to 1, poly2rc normalizes the prediction polynomial by a(1). k is a row vector of size length(a)-1.

[k,r0] = poly2rc(a,efinal)

returns the zero-lag autocorrelation, r0, based on the final prediction error, efinal.

A simple, fast way to check if a has all of its roots inside the unit circle is to check if each of the elements of k has magnitude less than 1:

stable = all(abs(poly2rc(a))<1)

Example

Consider an IIR filter given by

a = [1.0000   0.6149   0.9899   0.0000   0.0031  -0.0082];

Its reflection coefficient representation is

k = poly2rc(a)
k =
    0.3090
    0.9801
    0.0031
    0.0081
   -0.0082

Limitations

If abs(k(i)) == 1 for any i, finding the reflection coefficients is an ill-conditioned problem. poly2rc will return some NaNs and provide a warning message in this case.

Algorithm

poly2rc implements the recursive relationship:

This relationship is based on Levinson's recursion [1]. To implement it, poly2rc loops through a in reverse order after discarding its first element. For each loop iteration i, the function

1.: sets k(i) equal to a(i)

2.: applies the second relationship above to elements 1 through i of the vector a:

See Also


`ac2rc`	Conversion of autocorrelation sequence to reflection coefficients.
`latc2tf`	Lattice filter to transfer function conversion.
`latcfilt`	Lattice and lattice-ladder filter implementation.
`poly2ac`	Conversion of prediction polynomial to autocorrelation sequence.
`rc2poly`	Conversion of reflection coefficients to prediction polynomial.
`tf2latc`	Conversion of transfer function to lattice filter.

References

[1] Kay, S.M. Modern Spectral Estimation: Theory and Application. Englewood Cliffs, NJ: Prentice Hall, 1988.

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